A231654 -id:A231654 - cp's OEIS frontend

A231653 Number of non-equivalent ways to choose 4 points in an equilateral triangle grid of side n.

0, 0, 4, 41, 244, 1029, 3485, 9926, 25030, 57126, 120570, 238330, 446344, 797825, 1370684, 2274259, 3660612, 5734776, 8771181, 13127940, 19270240, 27789713, 39435814, 55142010, 76066910, 103627784, 139554142, 185929971, 245260890, 320527585, 415268815

Offset: 1

Heinrich Ludwig, Nov 12 2013

			For n = 3 there are the following a(3) = 4 choices of 4 points (=X) (rotations and reflections ignored):
    X        .        X        X
   X X      X X      X X      . .
  . X .    X . X    . . X    X X X

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1).

Cf. A234249, A001399, A227327, A230723, A231654, A231655.

a(n) = (n^8 + 4*n^7 - 6*n^6 - 32*n^5 + 84*n^4 - 32*n^3 - 16*n^2 - 192*n + B + C)/2304
where
  B = 84*n^3 - 234*n^2 + 168*n + 171   if n==1 (mod 2)
    = 0                                otherwise
and
  C = 128*n^2 + 128*n - 256            if n==1 (mod 3)
    = 0                                otherwise
G.f.: -x^3*(x^14 +7*x^12 +26*x^11 +146*x^10 +432*x^9 +947*x^8 +1418*x^7 +1621*x^6 +1405*x^5 +932*x^4 +438*x^3 +150*x^2 +33*x +4) / ((x -1)^9*(x +1)^4*(x^2 +x +1)^3). - Colin Barker, Feb 15 2014

A231655 Triangle T(n, k) read by rows giving number of non-equivalent ways to choose k points in an equilateral triangle grid of side n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 4, 2, 1, 1, 3, 10, 25, 41, 48, 41, 25, 10, 3, 1, 1, 4, 22, 87, 244, 526, 870, 1110, 1110, 870, 526, 244, 87, 22, 4, 1, 1, 5, 41, 238, 1029, 3450, 9147, 19524, 34104, 49231, 59038, 59038, 49231, 34104, 19524, 9147, 3450, 1029, 238

Offset: 0

Heinrich Ludwig, Nov 14 2013

Number of orbits under dihedral group D_6 of order 6. - N. J. A. Sloane, Sep 12 2019

			Triangle T(n, k) is irregularly shaped: 0 <= k <= n*(n+1)/2+1. The first row corresponds to n = 1, the first column corresponds to k = 0. Rows are palindromic.
  1,  1;
  1,  1,  1,  1;
  1,  2,  4,  6,  4,  2,  1;
  1,  3, 10, 25, 41, 48, 41, 25, 10,  3,  1;
  ...
There are T(3, 2) = 4 nonisomorphic choices of 2 points (X) in an equilateral triangle grid of side 3:
      X       .       .       X
     . .     X X     . .     X .
    . X .   . . .   X . X   . . .

Heinrich Ludwig, Table of n, a(n) for n = 0..173

Columns k=1..5 are A001399, A227327, A230723, A231653, A231654.

Cf. A054252, A283113, A289709, A326611.

cp's OEIS Frontend

A231653 Number of non-equivalent ways to choose 4 points in an equilateral triangle grid of side n.

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